How to limit a bone between other two bones, for scale and position
I'm not entirely sure what you're asking, but I'll take a guess:

If you want A's base to be at B's base and A's tip to be at C's base, you might benefit from drivers.  

Unfortunately, the expressions to map from B and C to the proper pose for A would not be trivial.

Also, blender has some shortcomings in its dependency system that prevents intra-armature drivers from being calculated correctly in some circumstances.